Ake, 30, has struggled for minutes under Pep Guardiola so far this season, starting just one of his eight Premier League appearances.
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According to Spanish outlet Sport, City would be willing to sell the Dutchman for a fee of around £21 million with Barcelona interested.
The report adds that the deal would still be difficult for Barcelona to complete due to their ongoing financial issues.
Barcelona are working to sign a new central defender in January following Andreas Christensen’s ACL injury and Ronald Araujo’s mental health crisis.